Element in Left Coset iff Product with Inverse in Subgroup

Theorem

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let $y H$ denote the left coset of $H$ by $y$.

Then:

$x \in y H \iff x^{-1} y \in H$

Proof

\(\ds x\)

\(\in\)

\(\ds y H\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists h \in H: \, \)

\(\ds x\)

\(=\)

\(\ds y h\)

Definition of Left Coset

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists h \in H: \, \)

\(\ds x^{-1}\)

\(=\)

\(\ds h^{-1} y^{-1}\)

Inverse of Group Product

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists h \in H: \, \)

\(\ds x^{-1} y\)

\(=\)

\(\ds h^{-1}\)

Product with $y$ on the right

\(\ds \leadstoandfrom \ \ \)

\(\ds x^{-1} y\)

\(\in\)

\(\ds H\)

$H$ is a subgroup

$\blacksquare$

Also see

• Element in Right Coset iff Product with Inverse in Subgroup

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.6 \ \text {(1L)}$ Another approach to cosets
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $7 \ \text{(i)}$